Graphics Reference
In-Depth Information
Figure 10.9.
Comparing the tangent with non-tangent
lines.
p
where c
k
and c
e
are nonzero. The integer k is called the
intersection multiplicity of
C
and
L
at
p
and is denoted by i(
C
,
L
;
p
). The integer i(
C
,
L
;
p
) is the analog, in the
context of intersections, of the multiplicity of a root of a polynomial with the same
intuitive connotation. The integer k can vary from line to line, therefore, define the
multiplicity of
C
at
p
, denoted by m
p
(
C
), by
()
=
(
)
m
C
min
i
C,L;p
.
p
line
L
through
p
A line
L
is then
tangent
to
C
at
p
if i(
C
,
L
;
p
) > m
p
(
C
). What this says is that a tangent
line is a line that intersects the curve more often at
p
than other lines and hence in
fewer points elsewhere. This agrees with one intuition of tangent lines, namely, that
they are lines that intersect a curve in only one point whereas nearby lines through
the point intersect the curve in additional points. See Figure 10.9.
Finally, the multiplicity m
p
(
C
) defined geometrically above has an algebraic
interpretation.
Definition.
Given a polynomial f(X
1
,X
2
,...,X
n
), expand f about a point
p
=
(x
1
,x
2
,...,x
n
) in a finite sum of the form
i
i
i
Â
(
)
=
(
)
(
)
(
)
fX X
,
,...,
X
a
X
-
x
1
X
-
x
2
.
..
X
-
x
n
.
12
n
i i
...
i
1 1
2 2
nn
12
n
,,
...,
i
ii
12
n
The smallest degree of all the monomials appearing in the expansion above is called
the
order of f at
p
and is denoted by ord
p
(f).
It is easy to see that
()
=
()
.
m
ord
f
p
p
This concludes our overview. It should prepare the reader for the various defini-
tions in this section and subsequent ones, in particular Section 10.17, where inter-
sections of lines and planes with varieties are used to isolate important concepts. Right
now we start back at the beginning.
Assume that
C
is a plane curve in
P
2
(
C
). Our main goal is to analyze points of
C
, in particular, certain special points, the “singular” points. In the process we shall
also define tangent lines. Assume that
C
is defined by the homogeneous polynomial
F(X,Y,Z) and consider a point of
C
. By Theorem 10.5.14 we can choose a coordinate
system in which the point does not lie on Z = 0 and in which Z = 0 is not a compo-
nent of
C
. With this choice of coordinate system we can study properties of
C
at our
